What the hell has it got to do with your mother, if you need some time to clear your head (and don’t we all), childcare is a perfectly reasonable option.
In terms of avoiding burning out, it sounds like you’re already there tbh. You really need to prioritise caring for yourself. You wouldn’t let your daughter be dehydrated or constipated so why is it ok for you? If your health slips you won’t be able to care for anyone so you need to start putting yourself towards the top of your list.
What childcare do you have when you’re at work? Can that be extended even by half a day to let you rest? Notice I didn’t say to let you clean, change beds, Hoover or cook - rest.
What’s your evening routine like? I tend to work hard to get everything done by bedtime (8.00pm for my two) and then stop. I might have a bath or watch tv or even go to bed with a glass of wine and a book but I stop, no housework, no laundry, if I can’t do it before 8.00 it waits. I usually get up at 6.15, kids up around 7. I use those 45 mins to pick up anything left from the night before and if possible a quick cup of tea.
Don’t be afraid to use screens to occupy her time while you do chores, so that when she’s in bed you can rest and take care of yourself. Does she have friends? I’m good friends with my DS best friends mum and we help each other out, even if it’s us having a coffee and a rant while the kids play.
I’m not a single mum, I have endless respect for folk doing this on their own, but I have had to really think about my stress and anxiety levels as a parent and these things work for me regardless of whether my OH is around to help.